A question involving a chess piece called a prince on the \(8\times 8\) chessboard leads to a concept in graph theory involving total domination in the Cartesian products of paths and cycles. A vertex \(u\) in a graph \(G\) totally dominates a vertex \(v\) if \(v\) is adjacent to \(u\). A subset \(S\) of the vertex set of a graph \(G\) is a total dominating set for \(G\) if every vertex of \(G\) is totally dominated by at least one vertex of \(S\). If \(S\) is a total dominating set of a graph \(G\), then \(S(v)\) denotes the number of vertices in \(S\) that totally dominate \(v\). A total dominating set \(S\) in a graph \(G\) is called a proper total dominating set if \(S(u) \ne S(v)\) for every two adjacent vertices \(u\) and \(v\) of \(G\). It is shown that \(C_n \ \Box \ K_2\) possesses a proper total dominating set if and only if \(n\ge 4\) is even and the graph \(C_n \ \Box \ P_m\) possesses a proper total dominating set for every even integer \(n \ge 4\) and every integer \(m \ge 3\). Furthermore, \(C_3 \ \Box \ P_m\) possesses a proper total dominating set if and only if \(m = 3\). If \(n\ge 5\) is an odd integer and \(m\equiv 3 \pmod 4\), then \(C_n \ \Box \ P_m\) has a proper total dominating set. If at least one of \(n\) and \(m\) is even, then \(C_n \ \Box \ C_m\) has a proper total dominating set. The graphs \(C_n \ \Box \ C_m\) are further studied when both \(n\) and \(m\) are odd. Other results and questions are also presented.

For a graph \(F\) and a positive integer \(t\), the vertex-disjoint Ramsey number \(VR_t(F)\) is the minimum positive integer \(n\) such that every red-blue coloring of the edges of the complete graph \(K_n\) of order \(n\) results in \(t\) pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to \(F\), while the edge-disjoint Ramsey number \(ER_t(F)\) is the corresponding number for edge-disjoint subgraphs. These numbers have been investigated for the three connected graphs \(K_3\), \(P_4\) and \(K_{1, 3}\) of size 3. For two vertex-disjoint graphs \(G\) and \(H\), let \(G+H\) denote the union of \(G\) and \(H\). Here we study these numbers for the two disconnected graphs \(3K_2\) and \(P_3+P_2\) of size 3. It is shown that \(VR_t(3K_2)= 6t+2\) and \(VR_t(P_3+P_2)= 5t+1\) for every positive integer \(t\). The numbers \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) are determined for \(t \le 4\) and bounds are established for \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) when \(t \ge 5\). Other results and problems are presented as well.

Every red-blue coloring of the edges of a graph \(G\) results in a sequence \(G_1\), \(G_2\), \(\ldots\), \(G_{\ell}\) of pairwise edge-disjoint monochromatic subgraphs \(G_i\) (\(1 \le i \le \ell\)) of size \(i\) such that \(G_i\) is isomorphic to a subgraph of \(G_{i+1}\) for \(1 \le i \le \ell-1\). Such a sequence is called a Ramsey chain in \(G\) and \(AR_c(G)\) is the maximum length of a Ramsey chain in \(G\) with respect to a red-blue coloring \(c\). The Ramsey index \(AR(G)\) of \(G\) is the minimum value of \(AR_c(G)\) among all red-blue colorings \(c\) of \(G\). Several results giving the Ramsey indexes of graphs are surveyed. A galaxy is a graph each of whose components is a star. Results and conjectures on Ramsey indexes of galaxies are presented.

Given a network modeled as a graph, a detection system is a subset of vertices equipped with “detectors” that can uniquely identify an “intruder” anywhere in the graph. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). In an OLD set, each vertex has a unique, non-empty set of detectors in its open neighborhood; meanwhile, in an IC, each vertex has a unique, non-empty set of detectors in its closed neighborhood. We explore one of their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector retains the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king grid, and presents the proof for the bounds on their minimum densities; \(\left[\frac{3}{10}, \frac{1}{3}\right]\) for RED:OLD sets and \(\left[\frac{3}{11}, \frac{1}{3}\right]\) for RED:ICs.

Exploring the vulnerability of any real-life network helps designers understand how strongly components or elements of the network are connected and how well they can function if there is any disruption. Any chemical structure can also be considered as a network in which the atoms correspond to the vertices, and the chemical bonds between the atoms correspond to the edges. Let \(G=(V, E)\) represent any simple graph with vertex set \(V\) and edge set \(E\). The vulnerability measure used in this paper is the paired domination integrity, defined as the minimum of the sum of any paired dominating set \(S\) of a graph \(G\) and the order of the largest component in the induced subgraph of \(V-S\). The minimum is found by considering all possible paired dominating sets of \(G\). In this paper, we obtain the paired domination integrity of the comb product of paths and cycles. In addition, we extend the study of graph vulnerability to chemical structures.

Let \(k, b, n\) be positive integers such that \(b\geq 2\). Denote by \(S(k,b,n)\) the numerical semigroup generated by \(\left\{b^{k+n+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\right\}\). In this paper, we give formulas for computing the embedding dimension and the Frobenius number of \(S(k,b,n)\).

Given a connected graph \(G=(V,E)\) of order \(n\ge 2\) and two distinct vertices \(u,v\in V(G)\), consider two operations on \(G\): the \(k\)-multisubdivision and the \(k\)-path addition. Let \(msd_{\gamma_c}(G)\) and \(pa_{\gamma_c}(G)\) denote, respectively, the connected domination multisubdivision and path addition numbers of \(G\). In both operations, \(k\) represents the number of vertices added to \(V(G)\), resulting in a new graph denoted by \(G_{u,v,k}\). We prove that \(\gamma_c(G) \le \gamma_c(G_{u,v,k})\) for \(k = msd_{\gamma_c}(G) \in \{1,2,3\}\) in the case of \(k\)-multisubdivision, where \(uv \in E(G)\). Additionally, we show that \(\gamma_c(G) – 2 \le \gamma_c(G_{u,v,k})\) for \(k = pa_{\gamma_c}(G) \in \{0,1,2,3\}\) in the case of \(k\)-path addition, where \(uv \notin E(G)\), and provide both necessary and sufficient conditions under which these inequalities hold.

This paper introduces two novel sequences: the \(k-\)-division Fibonacci–Pell polynomials and the \(k-\)-division Gaussian Fibonacci–Pell polynomials. Building on the well-known Fibonacci and Pell sequences, these new sequences are defined using a division-based approach, enhancing their combinatorial and algebraic properties. We present explicit recurrence relations, generating functions, combinatorial identities, and Binet-type formulas for these sequences. A significant contribution of the study is the factorization of the Pascal matrix via the Riordan group method using the proposed polynomials. Two distinct factorizations are derived, highlighting the algebraic structure and combinatorial interpretations of the \(k-\)-division polynomials. The work not only generalizes known polynomial sequences but also provides new insights into their matrix representations and applications.

This paper provides new lower bounds for van der Waerden numbers using Rabung’s method, which colors based on the discrete logarithm modulo some prime. Through a distributed computing project with 500 volunteers over one year, we checked all primes up to 950 million, compared to 27 million in previous work. We point to evidence that the van der Waerden number for \(r\) colors and progression length \(k\) is roughly \(r^k\).

Lightweight design is widely used for optimizing machinery. This paper proposes an algebraic–topology–based structural optimization method that formulates a mathematical model for mechanical equipment. The model is solved via sequential linear programming and recast as a function-optimization problem addressed by a Memetic algorithm combining a Newtonian local search, adaptive multipoint crossover, and stochastic variability. Using a bridge crane main girder as a case study, we minimize cross-sectional area with selected sectional parameters as design variables and constraints from the Crane Design Manual. With population size 100 and 300 maximum iterations, the optimized girder achieves a 13% weight reduction. Static and dynamic analyses confirm that strength and stiffness satisfy safe working conditions.